It works about the same way for all 4 games. It might seem complicated but is actually very easy to do.
NOTE: It does NOT work if you did something that would normally also wipe your save, such as deleting browser data or clearing cookies!
Here are the 7 steps to rescue your save on PC (for mobile, see the section below):
1. Copy the "command" for the game from its section below (the entire part in the green box. If you only copy parts of it, it does not work)
2. Open the game's old website
3. Open the console (F12 in Firefox, F12 then click Console in Chrome and Edge)
4. Paste the "command" for the game you're trying to get the save from (found below in the respective game's section)
5. Press enter and close the console (clicking an X button or pressing F12, usually)
6. Copy the (rescue) save appended at the bottom of the site entirely and store it somewhere (such as a .txt file)
7. Export it in the new hosted website, according to the short instructions described below. Done!
Since mobile has no console by default, it needs to be done a bit differently.
With liriliri's eruda program, it is possible to access the console on mobile, without any downloads or rooting required!
1. Create a new bookmark. The name does not matter
2. Set the address to the following, then save the bookmark:
3. Do steps 1-2 from the normal instructions (above)
4. Open the bookmark we have just created. This should add a little button in the bottom right. Click it to open the console
(Note: In Chrome you have to enter the bookmark's name in the search bar, instead of clicking it in the list of bookmarks)
5. Do steps 4-7 from the normal instructions (above). Done!